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VARIETIES OF COMPLETELY DECOMPOSABLE FORMS AND THEIR SECANTS
 

Summary: VARIETIES OF COMPLETELY DECOMPOSABLE FORMS AND
THEIR SECANTS
HIROTACHI ABO
Abstract. This paper is devoted to the study of higher secant varieties of va-
rieties of completely decomposable forms. The main goal is to develop methods
to inductively verify the non-defectivity of such secant varieties. As an appli-
cation of these methods, we will establish the existence of large families of
non-defective secant varieties of "small" varieties of completely decomposable
forms.
1. Introduction
In 1770, E. Waring suggested the problem of finding a positive integer s such
that every positive integer can be written as the sum of at most s dth
power of
positive integers. In 1909, this problem was solved affirmatively by Hilbert. There
is a polynomial version of this problem, which asks, "What is the smallest integer
s such that a general d-form in (n + 1) variables is expressible as the sum of s
dth
powers of linear forms?" This problem is often called Waring's problem for
polynomials. Waring's problem for polynomials has remained unsolved for many
years, but was completed in a series of papers [3, 2, 1] by Alexander and Hirschowitz

  

Source: Abo, Hirotachi - Department of Mathematics, University of Idaho

 

Collections: Mathematics